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ORIGINAL RESEARCH ARTICLE
published: 30 August 2012
doi: 10.3389/fncom.2012.00066
Fusimotor control of spindle sensitivity regulates central
and peripheral coding of joint angles
Ning Lan1,2*and Xin He1
1School of Biomedical Engineering, Med-X Research Institute, Shanghai Jiao Tong University, Shanghai, China
2School of Dentistry, Division of Biokinesiology and Physical Therapy, University of Southern California, Los Angeles, CA, USA
Edited by:
Hava T. Siegelmann, University of
Massachusetts Amherst, USA
Reviewed by:
Meng Hu, Drexel University, USA
Dimitri Nowicki, Moscow Institute of
Physics and Technology, Ukraine
*Correspondence:
Ning Lan, School of Biomedical
Engineering, Med-X Research
Institute, Shanghai Jiao Tong
University, 1954 Hua Shan Road,
Shanghai 200030, China.
e-mail: ninglan@sjtu.edu.cn
Proprioceptive afferents from muscle spindles encode information about peripheral joint
movements for the central nervous system (CNS). The sensitivity of muscle spindle is
nonlinearly dependent on the activation of gamma (γ) motoneurons in the spinal cord that
receives inputs from the motor cortex. How fusimotor control of spindle sensitivity affects
proprioceptive coding of joint position is not clear. Furthermore, what information is carried
in the fusimotor signal from the motor cortex to the muscle spindle is largely unknown. In
this study, we addressed the issue of communication between the central and peripheral
sensorimotor systems using a computational approach based on the virtual arm (VA)
model. In simulation experiments within the operational range of joint movements, the
gamma static commands (γs) to the spindles of both mono-articular and bi-articular
muscles were hypothesized (1) to remain constant, (2) to be modulated with joint angles
linearly, and (3) to be modulated with joint angles nonlinearly. Simulation results revealed a
nonlinear landscape of Ia afferent with respect to both γsactivation and joint angle. Among
the three hypotheses, the constant and linear strategies did not yield Ia responses that
matched the experimental data, and therefore, were rejected as plausible strategies of
spindle sensitivity control. However, if γscommands were quadratically modulated with
joint angles, a robust linear relation between Ia afferents and joint angles could be obtained
in both mono-articular and bi-articular muscles. With the quadratic strategy of spindle
sensitivity control, γscommands may serve as the CNS outputs that inform the periphery
of central coding of joint angles. The results suggest that the information of joint angles
may be communicated between the CNS and muscles via the descending γsefferent and
Ia afferent signals.
Keywords: muscle spindle, γscontrol, spindle sensitivity, Ia afferents, joint angle, central and peripheral coding
INTRODUCTION
Muscle spindle is a unique sensory organ that has dual efferent
and afferent innervations (Boyd, 1980; Matthews, 1981; Hulliger,
1984). A large amount of cortical outputs is directed to γ
motoneurons that supply fusimotor control of spindles (Boyd and
Smith, 1984). A larger number of studies have been dedicated
to elucidate the morphological, biochemical, and neurophysio-
logical properties of the spindle (Matthews, 1962; Granit, 1970;
Boyd and Smith, 1984). But relatively little has been revealed
about the functional role of fusimotor efferent in the execution
of motor tasks, because it has been difficult, if not impossible,
to record directly from gamma motor neurons during normal
movements. Fusimotor control is so far best understood to adjust
the sensitivity of muscle spindles. As the alpha motor neurons
activate extrafusal muscle fibers to produce a contraction force,
the spindle is unloaded. To keep the spindle sensitive during
muscle contraction, the central nervous system (CNS) may co-
activate the intrafusal fiber via descending gamma commands γ
(Vallbo and al-Falahe, 1990), in order to assess the outcome of
the alpha activation of muscles. In so doing, if γscommand were
properly modulated with movement, the spindle firing may not
be interrupted by the unloading effects of muscle contraction.
Early studies have associated the spindle function to regulation
of muscle length (Merton, 1953; Stein, 1974; Houk and Rymer,
1981). But difficulties of the length-servo hypothesis have turned
the direction of research towards more centrally organized pro-
gramming for motor control (Flash and Hogan, 1985; Feldman,
1986; Hasan, 1986; Corcos et al., 1989; Gottlieb et al., 1989; Uno
et al., 1989; Harris and Wolpert, 1998; Todorov and Jordan, 2002).
On the other hand, central programming or coding of sensori-
motor control must take into account the peripheral constraints
presented in the neuromuscular system (Kawato et al., 1990; Lan
and Crago, 1994; Lan, 1997). Thus, it is necessary to elucidate
the nature of information communicated between the central and
peripheral systems.
It has been a main subject of experimental studies with
regard to the nature of gamma fusimotor commands relevant
to motor control (Boyd, 1980; Matthews, 1981; Hulliger, 1984;
Boyd et al., 1985). Only until recently, experimental studies of
patterns of gamma motor activity during movement and pos-
ture in animals have shed some light to the plausible function of
fusimotor co-activation with αcommands (Taylor et al., 2004).
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COMPUTATIONAL NEUROSCIENC
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Lan and He Central and peripheral coding of joint angles
Direct recordings from gamma fibers in reduced cat preparations
showed that there was in-phase modulation of γsactivities with
muscle EMGs during locomotion, providing firm evidence of α–
γco-activation during movement (Taylor et al., 2000). And static
gamma activity was considered to be a fusimotor template of
intended movement (Taylor et al., 2006). This implied that γssig-
nal might carry centrally planned kinematic information of joint
angles. In the periphery, direct recording of Ia afferents from the
dorsal ganglion cells of decerebrated cats indicated a robust lin-
ear relation between Ia afferents and joint angles (Stein et al.,
2004). In human subjects, direct recording of spindle afferents
from the extensor carpi radialis brevis (ECRb) and extensor dig-
itorum (ED) (Cordo et al., 2002) revealed that the steady-state
population firing of Ia afferents was found linearly related to joint
position during the hold period between ramps. In these experi-
ments, γsmodulation of spindle sensitivity was unknown in both
animal and human recordings. However, the evidence in reduced
animal preparations and intact human subjects provided partial
clues on the central and peripheral coding of joint positions by
fusimotor (γs) commands and Ia afferent signals.
In a more theoretical approach, a number of studies have sug-
gested that trajectory and final position of movement may be
planned separately, and executed with a dual control strategy
(Lan et al., 2005; Ghez et al., 2007; Scheidt and Ghez, 2007).
Experimental evidence also indicated that the brain treats move-
ment and position information with distinct neural representa-
tions (Kurtzer et al., 2005). Injection of the γ-aminobutyric acid
(GABA) antagonist picrotoxin into cat’s reticular part of the sub-
stantia nigra (SNR) removed static fusimotor action from spindle
primary endings (Wand and Schwarz, 1985). On the other hand,
electrical stimulation at fasciculus retroflexus region of the cat’s
midbrain reproduced dynamic fusimotor effect, indicating that
the habenulo-interpeduncular system may be involved in generat-
ing dynamic gamma commands (Taylor and Donga, 1989). Thus,
movement and position control signals may be generated and
processed in different regions of the brain, and passed down to
spinal motor neurons as separate descending commands (Lemon,
2008). A set of static commands may be most relevant to main-
taining a steady state limb position (Lan et al., 2005), while a
set of dynamic commands may control dynamic acceleration and
deceleration of movements (Lan and Crago, 1994; Lan, 1997;
Lan et al., 2005). In this framework of dual control, it is neces-
sary that the CNS inform the peripheral neuromuscular system
about the desired joint position via a pathway separate from the
αcommands to the muscles. Recent experimental data (Cordo
et al., 2002; Taylor et al., 2006) imply that an alternative pathway
for transmission of kinematic information is via γcommands to
muscle spindles.
In this study, we used a computational approach to explore the
functional role of fusimotor system in transmitting the centrally
planned joint kinematics to the periphery, and how a robust lin-
ear relation between Ia afferent and joint angle could be achieved
with fusimotor control of spindle sensitivity. With a computa-
tional model of the virtual arm (VA) (Song et al., 2008a; He et al.,
2012), we tested three plausible strategies of fusimotor control
of spindle sensitivity with constant, linear and nonlinear mod-
ulations with joint angles. The correlation between joint angles
and Ia afferents under different fusimotor control strategies were
investigated for mono-articular and bi-articular muscles. The
hypotheses were rejected or accepted based on the consistence of
simulated behaviors to those of experiments. Part of the prelim-
inary results was presented in a conference proceeding (He and
Lan, 2011).
MATERIALS AND METHODS
THE SENSORIMOTOR SYSTEMS MODEL
The computational model of the integrated, multi-joint sensori-
motorVAsystemusedinthisstudywasshowninFigure 1.This
model has been developed and validated for simulation studies of
neural control of human arm movements (Song et al., 2008a; He
et al., 2012). The VA model was capable of generating Ia afferents
of muscles at different joint angles and under different fusimotor
inputs. Thus, it was suitable to address the issue of how fusimotor
control affects the coding of joint angles by Ia afferents. For com-
pleteness, a succinct description of the systems model was given
below.
The VA systems model in Figure 1 was a two-joint arm in
the horizontal plane. It consisted of subcomponent models of
an anatomically accurate upper arm with shoulder and elbow
joints, and physiologically realistic muscles and proprioceptors.
Each model component has been validated respectively during
its development (Cheng et al., 2000; Mileusnic et al., 2006; Song
et al., 2008a,b), and then integrated into the realistic VA systems
model in SIMULINK (Figure 1).
Computational modules of the VA model were implemented
with a graphic modeling software SIMM and SIMULINK, respec-
tively. The mathematical equations of geometry, kinematic, and
dynamics of the multi-body system of the upper arm were
embodied into SIMM, and the SIMM model was converted into a
computational block in SIMULINK that computed joint motion
with given muscular forces acting upon the joints (Song et al.,
2008a). There were six representative muscles acting on the joints.
The virtual muscle (VM)model contained all mathematical equa-
tions that described realistic muscle physiology and mechanics
(Cheng et al., 2000), and a new version of the VM model was
implemented in SIMULINK (Song et al., 2008b). The VM mod-
ule computed muscle force and muscle fascicle length with given
neural input after a continuous recruitment scheme (Song et al.,
2008b). The new VM model improved computational efficiency
and simulation stability. It allowed a continuous recruitment of
slow and fast fibers, and decoupled α,γcommand inputs to active
extrafusal and intrafusal fibers, respectively.
Three pairs of agonist and antagonist muscles were selected to
actuate two degrees of freedom (DOF) of the VA model in hor-
izontal plane (Figure 1). Pectoralis major (clavicle portion, PC)
and Deltoid posterior (DP) were mono-articular flexor and exten-
sor at the shoulder joint; brachialis (BS) and triceps brachii lateral
head (Tlt) were mono-articular flexor and extensor of the elbow
joint; biceps brachii short head (Bsh) and triceps brachii long
head (Tlh) were the bi-articular muscles cross both joints.
Each muscle model was embedded with a spindle model
(Mileusnic et al., 2006) and a simplified Galgi tendon organ
(GTO) model (Song et al., 2008a). The spindle model contained
a bag1, a bag2, and a chain fiber with Ia and II afferent outputs.
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Lan and He Central and peripheral coding of joint angles
FIGURE 1 | The virtual arm (VA) model is an integrated neuromuscular
sensorimotor systems model in SIMULINK, which encompasses an
anatomically accurate structure of upper arm, physiologically realistic
muscle mechanics and dynamics, and spindle and Golgi tendon organ
(GTO) proprioceptors. Each subcomponent embodies a set of mathematical
equations obtained from previous experimental data in literature that
describe the physiological, geometrical, kinematic, and dynamic properties of
the subsystems. The VA model receives αand γcommands from the central
nervous system (CNS), and outputs numerical results of simulation for all
state variables, including joint kinematics and proprioceptive afferents (i.e., Ia,
Ib,andII afferents). The biomechanical model of the VA has two degrees of
freedom (DOFs) in horizontal plane (shoulder flexion/extension, elbow
flexion/extension) and is driven by six muscles, which are clavicle portion of
pectorailis major (PC) and deltoid posterior (DP) for shoulder joint, brachialis
(BS) and triceps lateral head (Tlt) for elbow joint, and biceps short head (Bsh)
and tricps long head (Tlh) cross both joints. The virtual muscle (VM) model
activated by commands calculates contraction forces (Fm)and instantaneous
muscle fascicle length (Lce). The muscle spindle model receives inputs of
fascicle length (Lce) and fusimotor modulation (γs,γd) and generates primary
(la) and secondary (II) afferents. However, since we are interested in neural
coding for joint angles in this study, only γsand Ia afferent signal is of interest
in the simulation and analysis.
Ia afferents were sum of all fiber outputs and II afferents were pri-
marily from chain fibers. The gamma static efferent innervated
bag2 and chain fibers, and the gamma dynamic efferent inner-
vated primarily bag1 fiber. Thus, the spindle model was capable of
simulating spindle Ia and II responses to both static and dynamic
fusimotor inputs.
DETERMINATION OF A SET OF EQUILIBRIUM POSITIONS IN
SIMULATION
The definitions of shoulder and elbow angles were shown in
Figure 2A. The range of shoulder flexion was set from 0◦(fully
extended) to 120◦(fully flexed), and the range of elbow flex-
ion was from 0◦(fully extended) to 150◦(fully flexed). Showing
in Figure 2A were a typical mono-articular muscle crossing the
elbow joint, and a typical bi-articular muscle crossing both shoul-
der and elbow. The spindles were arranged in parallel with muscle
fascicle fibers. In this study, however, joint angles of shoulder and
elbow were varied in the operational range within the full range
of motion (ROM), as shown in Figure 2B.
A procedure of initialization for dynamic simulation used in
(He et al., 2012) was adopted in this study to obtain a set of equi-
librium positions as shown in Figure 2B.Theαcommands of
the nine stable equilibrium positions were tabulated in Tab l e 1.
The procedure was effective to determine initial system parame-
ters, such as, fascicle length and joint angles, so that simulation
could converge and the shoulder and elbow joints could be stabi-
lized to a desired equilibrium position. In each simulation, the
total running time was about 30 (s), in which the initial 10 s
were designed to allow simulation to converge. A random, sig-
nal dependent noise (SDN) (Jones et al., 2002) was added to the
muscle activation (He et al., 2012)atabout10(s)toreproduce
the inherent variability in the neuromuscular system. The steady
state joint angles and Ia afferents were calculated as the average
value of data in the last 10 s of simulation.
At the set of equilibrium positions, the geometric relation-
ships between joint angle and muscle fascicle length in all muscles
was evaluated. This was one of the peripheral constraints for the
central programming of control of both intrafusal and extra-
fusal fibers. The joints of the VA were placed to different angles
in the workspace (Figure 2B) by choosing particular patterns of
mono-articular muscle activations (Ta b l e 1). The musculotendon
lengths of the muscles were calculated from the VA model and the
corresponding fascicle lengths were obtained at these joint angles.
The relationship between joint angles and muscle fascicle length
in the operational range of joint movement was then assessed in
Figure 3.
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Lan and He Central and peripheral coding of joint angles
FIGURE 2 | (A) Geometric definition of shoulder and elbow angles. The
range of shoulder flexion is set from 0◦(fully extended) to 120◦(fully
flexed), and the range of elbow flexion is from 0◦(fully extended) to 150◦
(fully flexed). Showing in the figure are a typical mono-articular muscle
crossing the elbow joint, and a typical bi-articular muscle crossing both
shoulder and elbow joints. The spindles are arranged in parallel with muscle
fascicle fibers. We hypothesize that muscle fascicle length (Lce )andIa
afferent are related to the corresponding joint angles of span. Thus for
bi-articular muscles, they are related to the sum of joint angles of span.
(B) Nine sets of αstat commands (Ta b l e 1 ) are used to stabilize the VA
model at nine equilibrium positions (1 ∼9) in horizontal plane, respectively.
At each position the spindle sensitivity control by γstat is investigated.
EVALUATION OF SPINDLE SENSITIVITY
In this study, we focused on the effects of gamma static, γs,con-
trol of spindle sensitivity with respect to muscle fascicle length
change, while the gamma dynamic control was fixed to a con-
stant level. An example of influences of gamma static control
and joint angles on spindle sensitivity was explored in the six
muscles. First, at a fixed joint configuration, the gamma static
commands to all muscles were varied in a ramp pattern, and
the Ia afferents of the six spindles showed simultaneous vari-
ation with the ramp change of the gamma static command.
Then, the sensitivity of Ia afferent to fascicle length change was
examined in response to ramp changes in joint angles with con-
stant levels of gamma static inputs in the six muscles. These
results were shown in Figure 4,andtheyverifiedtheIa sensitivity
Table 1 | Alpha static activation levels at nine positions.
Muscle Position
123456789
PC 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65
DP 0.65 0.60 0.55 0.50 0.45 0.40 0.35 0.30 0.25
Bsh 000000000
Tlh 000000000
BS 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65
Tlt 0.65 0.60 0.55 0.50 0.45 0.40 0.35 0.30 0.25
to both gamma static control and joint angle (or fascicle
length).
The landscape of spindle Ia sensitivity with respect to joint
angles and gamma static control was then further evaluated. In
these simulations, alpha (αs) commands represented the activa-
tion level of motor neuron pool as inputs to the VA model. Each
set of constant motor commands (αs,γs) produced an equilib-
rium position of the arm with Ia afferents from six muscles. A
total of nine sets of alpha commands to the shoulder and elbow
muscles (Ta b l e 1 ) positioned the VA model to nine different equi-
librium angles in the shoulder and elbow joints (Figure 2B). The
gamma static (γs) commands were changed from 0.0 to 1.0 with
an increment of 0.1 at each of the joint angles. A total of 81
points in the θEP −γs−Ia space formed the landscape surface
of Ia sensitivity for each muscle (Figure 5), which revealed the
fundamental relationship among the three variables.
TEST OF FUSIMOTOR CONTROL STRATEGIES
Three sets of simulation experiments were designed to evalu-
ate the plausible strategies regarding spindle sensitivity control.
Based on the shape of the sensitivity landscape in the θEP −
γs−Ia space, fusimotor control strategies, represented by the
relation between equilibrium angle and gamma static command
(γs−θEP), were hypothesized (1) to remain constant for all joint
angles (H1); (2) to be modulated linearly with joint angles (H2);
and (3) to be modulated quadratically with joint angles (H3).
The resultant relation between Ia afferents and joint angles was
evaluated under the three hypotheses of fusimotor control, and
the outcome relation of the (Ia −θEP)curvewascomparedto
experimentally observed behaviors. The necessary condition to
reject a hypothesis was that the outcome relation of Ia afferents
with joint angles must be a linear relation (Cordo et al., 2002;
Stein et al., 2004). But to form a sufficient set of conditions to
accept a hypothesis, other physiological constraints of experimen-
tal evidence in addition to the linear (Ia −θEP ) output must be
considered.
RESULTS
THE LENGTH-ANGLE RELATION OF MUSCLES
The relation between muscle fascicle length and joint angle
revealed a geometric constraint in the VA model. The geomet-
ric relation was characterized by the θEP −Lce curves shown in
Figure 3 for the six muscles. In the operational range of shoulder
and elbow joints, simulation results showed that the fascicle
Frontiers in Computational Neuroscience www.frontiersin.org August 2012 | Volume 6 | Article 66 |4
Lan and He Central and peripheral coding of joint angles
FIGURE 3 | Relation between equilibrium joint angle and muscle
fascicle length (θEP −Lce )of each muscle obtained in the range of joint
angles used in simulation. (A) Relation of shoulder mono-articular
muscles PC and DP.(B) Relation of elbow mono-articular muscles BS and
Tlt. (C) Relation of bi-articular muscles Bsh and Tlh cross both shoulder and
elbow joints. Results indicate that a nearly linear relation exists between
muscle fibre length and joint angle for both mono-articular muscles and
bi-articular muscles, because of the their arrangement. The fascicle length
of flexor is shortened and that of extensor is lengthened with increase of
the joint angles of span. For bi-articular muscles Bsh and Tlh, their fascicle
length, Lce, is found linearly related to the sum of shoulder and elbow
angles. The linearity in the geometric relations provides supportive
evidence for a simple coding relationship between joint angles and spindle
input and output that are related to muscle fascicle length.
length of both mono-articular and bi-articular muscles was lin-
early related to the joint angles they cross. For mono-articular
muscles, the joint of span was either shoulder joint or elbow
joint, and thus the fascicle length of mono-articular muscles was
linearly proportional to either shoulder angle or elbow angle
FIGURE 4 | Responses of primary afferents (Ia) of muscle spindles to
(A) ramped γsdrive, and (B) ramped joint angle (fascicle length)
change, respectively. (A) The VA was maintained at position 5, while γs
commands of flexors ramped from 0.3 up to 0.8 and those of extensors
ramped from 0.7 down to 0.2, concurrently within 5 s. The Ia afferents of all
muscles were shown to be modulated in-phase with γschanges. (B) The
VA was moved from position 4 to 6 within 5 s by ramped alpha commands
of single joint muscles, while the γscommands of each muscle remained
constant at 0.5. The muscle fascicle length changed simultaneously with
joint angles, and the Ia afferents were modulated in-phase with changes in
fascicle length Lce. These demonstrate the sensitivity Ia afferents with
respect to fusimotor activation γsand muscle fascicle length Lce.
Frontiers in Computational Neuroscience www.frontiersin.org August 2012 | Volume 6 | Article 66 |5
Lan and He Central and peripheral coding of joint angles
(Figures 3A,B). For bi-articular muscles, the joints of span were
both shoulder and elbow joints, and thus the fascicle length was
linearly proportional to the sum of shoulder and elbow angles
(Figure 3C). It is worth to note that while this fitted relation
was near linear within the operational range of joints, significant
nonlinearity may occur at the two extremes of ROM of joints,
because the wrap around curvature of the joint was most effec-
tive within the operational ROM. Thus, at the extreme of joint
angles, the CNS may rely on additional modality of proprio-
ception, such as joint receptors, to estimate the value of joints
accurately.
SPINDLE RESPONSES TO CHANGES IN GAMMA STATIC AND
FASCICLE LENGTH
Figure 4 illustrated that gamma fusimotor control and fascicle
length changes can modulate the sensitivity of Ia afferents effec-
tively. The primary (Ia) afferents of muscle spindles responded
to ramped gamma static drive (Figure 4A)andrampedfascicle
length (Figure 4B) differently. In the simulations for fusimotor
modulation effects, the VA was stabilized at position 5, and the
gamma static commands of flexors were ramped from 0.3 up to
0.8 and those of extensors from 0.7 down to 0.2 during a period
of 5 s. It was shown that the Ia afferents of all muscles were
modulated in proportion with gamma static changes (Figure 4A),
showing a strong modulation of Ia sensitivity by gamma static
commands. In the simulation for joint angle changes, the VA was
moved from position 4 to 6 during a period of 5s. Joint angles
were ramped from initial position to destination with linear
changes in alpha drives to single joint muscles, while gamma static
commands of each muscle remained constant at 0.5. Figure 4B
showed the response of the Ia afferents to joint angle (or mus-
cle fascicle length) changes. Clearly, there was a speed sensitivity
component in the Ia response that was not present in isomet-
ric gamma static sensitivity of Figure 4A.Afterreachingtothe
destination of joint position, the spindle outputs were generally
settled to a new level, but were affected by the dynamics in the
spindle model (Mileusnic et al., 2006), as well as noise in the neu-
romuscular system (Jones et al., 2002). In this case, the fascicle
length variation was about 10%, and the Ia outputs at steady state
were varied approximately with the similar percentage, indicating
effective modulation of Ia sensitivity by joint angles (or fascicle
length).
THE LANDSCAPE OF SPINDLE SENSI TIVITY
In order to obtain a general view of spindle Ia afferents in
the normal range of joint angles and with all possible gamma
static values, we searched the joint angle—fusimotor command—
primary afferent (θEP −γs−Ia) space. In these simulations, the
joint angles were fixed to one of the nine positions in Figure 2B.
Then the gamma static inputs to the six muscle spindles were
varied from 0.1 to 1.0, and Ia afferents of the six muscle spin-
dles were examined. The results of each muscle were plotted
in the 3D graphs in Figure 5. It was shown in general that the
response of Ia afferents was not linear with respect to either joint
angles or fusimotor commands in the whole space. This was not
surprise because of the nonlinear response of spindles to both
fusimotor commands and fascicle length change. Nonlinearity
mainly occurred at the shorter muscle length, where the intra-
fusal fibres were relaxed at lower gamma activation levels. On the
other hand, saturation of Ia afferents at longer muscle length with
higher gamma activation levels also gave rise to nonlinear spindle
sensitivity. The landscape of Ia sensitivity depicted the interrela-
tion between gamma static commands and joint angles that may
dictate the fusimotor control strategy.
PLAUSIBLE FUSIMOTOR CONTROL STRATEGIES
If joint angle is coded in the descending γscommands in the
CNS, the γsshould be formulated as a function of joint angle,
so that the resultant relation between Ia afferents and joint angles
matches to that of experimentally observed linear relation. Thus,
the central strategy of joint angle coding must take into account
of the spindle sensitivity with respect to joint angles and fusimo-
tor activation in Figure 5. Since the CNS has the luxury to control
spindle sensitivity by fusimotor commands, the CNS may modu-
late spindle sensitivity by adjusting the central pattern of coding
of joint angles, in order to keep the peripheral coding of joint
angle by Ia afferents consistent and reliable under all conditions.
The modulation of γscommand may occur in a relatively nar-
row region, for example a constant γsvalue; or in a wide range
of γscommand between 0.0 and 1.0 in order to achieve a better
resolution of coding. In this study, we examined three scenar-
ios (hypotheses) regarding fusimotor control strategy: (1) γswas
maintained constant within the range of joint angle; (2) γswas
modulated with joint angle in a linear function of joint angle; and
(3) γswas modulated in a nonlinear function with joint angles.
The first scenario was tested by postulating that γscommands
remained constant at modest activation levels within the entire
range of joint angle θ(Figures 6A–C). The primary afferents
of mono-articular muscle spindles showed a good linear rela-
tionship with corresponding joint angles (Figures 6D,E). The Ia
afferents of bi-articular muscle spindles were also linearly propor-
tional to the sum of shoulder and elbow joints as well (Figure 6F).
The Ia afferents of extensor DP, Tlt, and Tlh were positively pro-
portional to the joint angles, and Ia afferents of flexor PC, BS,
and Bsh were negatively proportional to joint angles (Ta b l e 2 ).
However, it was clear from Figure 5 that such good linear rela-
tions in θEP −Ia were only possible from median to high levels of
γsactivations. A constant fusimotor control may not be a physio-
logical strategy because this does not allow any central coding of
joint angles with γs. In addition, experimental evidence did not
support constant γscommands during movements (Taylo r e t a l.,
2000, 2004, 2006).
The second scenario was then examined with γsvaried lin-
early with joint angle θ(Figures 7A–C). The linear relation of
θEP −γscould be learned by the CNS to specify desired position
of joints. However, the results of Ia afferents of both mono-
articular and bi-articular muscle spindles were not linearly related
to joint angles, as was shown in Figures 7D–F.Thisnonlinear
response was evident from the nonlinear landscape of Ia sensitiv-
ity shown in Figure 5, where nonlinearity occurred in the lower
and higher regions of gamma activation and joint angles. As a
result, there was saturation in the Ia response as indicated by the
arrows in Figure 7. Thus, the outcome of the linear hypothesis
of fusimotor control strategy did not give rise to a linear relation
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Lan and He Central and peripheral coding of joint angles
FIGURE 5 | The landscape of Ia afferent sensitivity with respect to the
full ranges of fusimotor commands and muscle fascicle lengths
(θEP −γs−Ia). This is obtained by increasing fusimotor drive at each joint
angle for all muscles incrementally. The sensitivity landscapes clearly reveal
the complex interrelations of Ia afferents with both fusimotor control and joint
angles. In general, the interrelation is nonlinear, and the nonlinearity is more
prominent at the lower and higher values of fusimotor commands and joint
angles. These nonlinearities may be due to sluggish sensitivity in the short
fascicle length and saturation in the long fascicle length. However, it is also
clear that the nonlinearity exists in the middle range of fusimotor commands
for all muscles. This phenomenon reflects the nonlinear nature of
physiological responses of muscle spindles.
in θEP −Ia that was observed in experiments (Cordo et al., 2002;
Stein et al., 2004). Therefore, the hypothesis of linear central cod-
ing of joint angle by γscommandwasrejectedasaplausible
strategy of fusimotor control.
Consequently, this led us to consider the third scenario of a
nonlinear monotonic coding of joint angle by fusimotor com-
mand, which would yield a linear output in Ia afferents with
respect to joint angles. We hypothesized that γscommands were
modulated quandratically with joint angle θin parabolic curves of
θEP −γs,asshowninFigures 8A–C for the six muscles. Results
showed that the Ia afferents of the six muscles were well corre-
lated with joint angles linearly, as shown in Figures 8D–F,with
a goodness of fitting R2>0.99. The outcome of this strategy
appeared to fit all experimental data available (Cordo et al., 2002;
Stein et al., 2004), and thus it may be the most likely strategy
that the CNS may adopt for fusimotor control of spindle sensi-
tivity. Note that although the central coding relationship between
θEP −γsis non-linear, it remains a monotonic curve. This implies
that γscould encode joint position uniquely within the ROM of
joints. This phenomenon suggests that it is possible to manipulate
the fusimotor commands to linearize the nonlinear Ia sensitiv-
ity revealed in Figure 5. The coefficients of fitted equations are
presented in Ta b l e 3 , which may be used in future simulation
studies.
DISCUSSION
The understanding of neural control of movement would not
be complete without revelation of the nature of fusimotor con-
trol for intrafusal fibers. The presence of enormous efferent
and afferent innervations in the spindle prompts the question
of what functional implications this sophisticated system may
have, given that there is a vast CNS neural circuitry dedicated
to process efferent and afferent neural information. The affer-
ent signals may provide the CNS with peripheral kinematic
information that allows the CNS to assess the outcome of exe-
cuted motor action. But it is not straightforward as to what
Frontiers in Computational Neuroscience www.frontiersin.org August 2012 | Volume 6 | Article 66 |7
Lan and He Central and peripheral coding of joint angles
FIGURE 6 | The constant fusimotor control strategy, θEP −γs,curves
(A–C), and Ia afferents response (θEP −Ia) curves of muscle spindles (D–F),
for each muscle. The (A,D) is for shoulder actuators, the (B,E) is for elbow
actuators, and the (C,F) is for bi-articular muscles. The abscissas indicate joint
flexion angles while the vertical axis indicate activation levels (A–C) or firing
rates (D–F).Theγsinputs of muscle spindles are set to medium constant
levels at different angular positions, the firing rates of Ia afferents show an
excellent linear relation with joint angles (with a goodness of fitting R2>0.99).
This result may be evident from the sensitivity landscape of Figure 5, from
which a constant γsvalue results in a fairly linear (θEP −Ia) relation.
Table 2 | Fitting coefficients of constant γsstrategy.
γs=aθ2+bθ+cIa=kθ+e
abc k e R
2
PC 0 0 0.55 −1.4931 173.99 0.9994
DP 0 0 0.45 1.1893 45.083 0.9994
BS 0 0 0.65 −1.114 8 191 .58 0.9948
Tlt 0 0 0.35 0.9195 38.098 0.9974
Bsh 0 0 0.60 −0.8712 207.90 0.9921
Tlh 0 0 0.40 0.6615 7.6490 0.9998
Average – 1.0416 ±0.2644 0.9972
(pps/◦)
information content is carried in the efferent fusimotor signals
to the spindle in the periphery. A more compelling fact is that
there are more neurons in the motor cortex that innervate spinal
gamma motoneurons than those that control alpha motoneu-
rons (Boyd and Smith, 1984). Thus, it is imperative to under-
stand the significance of the large amount of corticospinal out-
flows of fusimotor control to the peripheral musculoskeletal
system.
There have been early efforts to identify indirectly the profiles
of fusimotor control signals during motor performance (Hulliger
and Prochazka, 1983; Hulliger et al., 1987). This was largely
conducted with Ia afferent data recorded from reduced prepara-
tions of passively behaving animal models. An inverse simulation
method was proposed to deduce the fusimotor profile from
recorded Ia afferents along with movements. However, it was then
realized that the fusimotor profile could be entirely different in
voluntarily behaving animals from those in reduced animal mod-
els. Also, there were intermediate variables, such muscle fascicle
length, musculotendon length and joint angle, that may all affect
the accuracy of estimates. A more stringent limit was due to
the nonlinear dynamics of the musculoskeletal responses, which
may lead to a non-unique pattern of fusimotor activity, even
though optimization technique may help reduce the uncertainty
of estimation.
In spite of technical difficulty, recently, direct recording from
gamma motor neurons in reduced animal models had success-
fully produced considerable insight into the fusimotor activation
profiles during movements (Taylor et al., 2000, 2004, 2006). It
was observed that static gamma activity formed a fusimotor tem-
plate of intended movement (Taylor et al., 2006), and may carry
kinematic information of joint angles. In the periphery, direct
recording of Ia afferents from the dorsal ganglion cells of decere-
brated cats suggested a similar conclusion that Ia afferents carried
joint angle information (Stein et al., 2004). Similar conclusion
was confirmed in human subjects with a voluntary contraction
task (Cordo et al., 2002), which revealed that the steady-state
population firing of Ia afferents was found linearly related to
joint position during the hold period between ramps. Notice
that in the experiments of Stein et al. (2004)andCordo et al.
(2002), the γsmodulation of spindle sensitivity was unknown.
But separate recordings of fusimotor activities and Ia afferents
suggested that fusimotor efferents may be programmed in the
CNS such that Ia afferents reliably inform the kinematics of
peripheral limb movements. More importantly, these experimen-
tal results formed a set of necessary conditions that simulated
behaviors of muscle spindle apparatus must conform. In the
sense of mathematical proof using computational methods, a
Frontiers in Computational Neuroscience www.frontiersin.org August 2012 | Volume 6 | Article 66 |8
Lan and He Central and peripheral coding of joint angles
FIGURE 7 | The linear fusimotor control strategy, θEP −γscurves (A–C),
and Ia afferents response, (θEP −Ia) curves of muscle spindles (D–F), for
all muscles. The (A,D) is for shoulder actuators, the (B,E) is for elbow
actuators, and the (C,F) is for bi-articular muscles. The axes were defined the
same as in Figure 6. When γsinputs of muscle spindles are
linearly modulated with joint angles from 0.3 to 0.9, the firing rates of Ia
afferents do not display a linear and monotonical relation to joint angles.
Saturations of Ia responses occur at each muscle, as indicated by the arrows.
This is due to the nonlinear physiological properties of muscle spindle
demonstrated in Figure 5, and would not be possible to avoid if the full
range of fusimotor commands were to be used to encode joint angle
information.
FIGURE 8 | The quadratic strategy of fusimotor control, θEP −γscurves
(A–C), and Ia afferents response, (θEP −Ia) curves of muscle spindles
(D–F), for each muscle. The (A,D) is for shoulder actuators, the (B,E) is for
elbow actuators, and the (C,F) is for bi-articular muscles. The axes were
defined the same as in Figure 6. When γs
modulated with joint angles θEP quadratically in a monotonical manner with
joint angles from 0.3 to 0.9, the firing rates of Ia afferents show a robust linear
relation with joint angles with an average goodness of linear fitting R2>0.99.
This phenomenon suggests that it is possible to manipulate the fusimotor
commands to linearize the nonlinear Ia sensitivity revealed in Figure 5.
hypothetical fusimotor control strategy must reproduce all exper-
imentally observed behaviors simultaneously. Strategies that do
not simultaneously satisfy these necessary conditions could not
be considered biologically plausible.
With this criterion in mind, we used a computational approach
to address these related issues, (1) what specific kinematic infor-
mation is encoded in gamma static fusimotor efferents? And
(2) how gamma static command may be controlled in order
Frontiers in Computational Neuroscience www.frontiersin.org August 2012 | Volume 6 | Article 66 |9
inputs of muscle spindles are
Lan and He Central and peripheral coding of joint angles
Table 3 | Fitting coefficients of quadratic γsstrategy.
γs=aθ2+bθ+cIa=kθ+e
ab c R
2keR
2
PC 6e-5 0.0020 0.3835 0.9996 −0.7757 145.48 0.9893
DP 7e-5 −0.0116 0.9595 0.9970 0.6500 86.787 0.9983
BS 6e-5 −0.0010 0 .3855 0.9992 −0.5195 145.16 0.9931
Tlt 2e-5 −0.0050 0.7772 0.9961 0.6469 91.982 0.9897
Bsh 2e-5 −0.0008 0.3698 0.9993 −0.4712 154.54 0.9939
Tlh 8e-6 −0.0044 0.9292 0.9988 0.3386 66.437 0.9966
Average – 0.9983 0.5670 ±0.1417 0.9935
(pps/◦)
to maintain a consistent Ia encoding of joint angles during
movements. The peripheral factors that may affect the out-
come of central coding were examined first with the VA model
in Figures 3–5. It was interesting to note that the variability
caused by the internal SDN noise simplified the relation between
muscle fascicle length and joint angles in certain degree to a
proximately linear relation within the operating range of joint
angles, because of the averaging effects as shown in Figure 3.This
appeared to lessen the nonlinear effects in the peripheral mus-
culoskeletal system. However, the spindle sensitivity did show
a significant nonlinear response that could affect both central
and peripheral coding of joint angles (Figure 5). With promi-
nent nonlinear spindle response, three hypotheses were tested
regarding the central coding strategies of joint angles. The first
set of simulation rejected the hypothesis that a constant γscon-
trol may be a plausible neural strategy in spite of its excellent
linear θEP −Ia relation. Experimental evidence clearly showed
that a dynamic pattern of γsmodulation was observed to co-
vary with αcommand and joint angles during movements (Ta y lor
et al., 2000, 2004, 2006). In the test of second hypothesis of
this study, the linear θEP −γsmodulation did not produce a
well-regulated linear θEP −Ia relation, which was observed in
experimental recordings in man and animals (Cordo et al., 2002;
Stein et al., 2004). This outcome may be attributable to the non-
linear sensitivity presented in the spindle response in Figure 5.
Thus, the hypothesis of linear control strategy of spindle sensi-
tivity expressed by linear θEP −γscorrelation was also rejected.
Then, we tested the third hypothesis, in which a nonlinear con-
trol strategy of γscommands may avoid the nonlinear zone
in the landscape of spindle sensitivity in Figure 5.Theresults
indicated that a second order nonlinear relation between γscom-
mand and joint angle, i.e., a parabolic θEP −γscurve, was indeed
necessary, in order for the Ia afferent to be linearly correlated
with joint angle. With the quadratic strategy, the Ia afferents
of both mono-articular and bi-articular muscles displayed the
similar property of a robust linear relation with joint angles.
This result implies that the brain could learn the peripheral
constraints, and program the nonlinear central coding of the
monotonic θEP −γscurve, and send the γscommand to inform
the periphery about the centrally planned joint angles. Such cen-
tral coding strategy would also allow the spindle to maintain
an accurate and consistent encoding of angular information in
Ia afferents, which the brain needs to evaluate the peripheral
performance.
Lastly, we calculated the average position sensitivity of Ia affer-
ents of all six muscles from simulation results for the three
strategies of fusimotor control. The slopes of the linear θEP −Ia
relationship were compared to that of Cordo’s et al. (2002)human
physiological recordings. The average position sensitivity under
constant gamma control was 1.04 ±0.29 pps/◦(Ta b l e 2), which
was much higher than that of quadratic strategy of gamma modu-
lation of 0.57 ±0.16 pps/◦(Ta b l e 3 ). The latter was closer to that
of position sensitivity of holding rate measured by Cordo et al.
(2002), which was 0.40 ±0.30 pps/◦. This further supports the
quadratic hypothesis as a plausible fusimotor control strategy for
postures and movements.
The implication that fusimotor signals encode kimematic
information of planned (or desired) movements is consistent with
the finding that fusimotor activities were enhanced when per-
forming a naïve task (Hulliger, 1984). When a new movement
is performed, the CNS needs to program an optimal pattern
of kinematics that is represented in fusimotor commands. This
may necessitates modifying the planned kinematics frequently
from practice to practice. The heightened activities seen in the
fusimotor signals suggest that a process of searching for optimal
kinematics is going on for the new task. In this process, pro-
prioceptive afferents are used to assess the outcome of motor
action, and are compared to the centrally programmed kinemat-
ics to detect any deviations between the programmed movement
and outcome movement. Modifications are made in both cen-
trally planned kinematics (gamma commands) and motor actions
(alpha commands) to further optimize movement performance.
Thus, the motor learning and control system acts like a reference
adaptive control system, where the gamma commands provide
the reference trajectories of movement (Taylor et al., 2006), and
the alpha commands produce optimal driving inputs to the extra-
fusal muscle fibers. To fully appreciate the reference adaptive
control of movement, central encoding of dynamic fusimotor
commands with respect to movement kinematics should also be
elucidated in the future research.
CONCLUSION
We examined the peripheral factors that may influence the cen-
tral and peripheral coding of joint angles through efferent and
Frontiers in Computational Neuroscience www.frontiersin.org August 2012 | Volume 6 | Article 66 |10
Lan and He Central and peripheral coding of joint angles
afferent innervations of the spindle apparatus. Based on these
peripheral constraints, we have tested three hypotheses regarding
static fusimotor control strategies of mono-articular and bi-
articular muscles to achieve a reliable encoding and decoding of
joint angle information. Results suggest that a quadratic strategy
of static fusimotor control could lead to a linear relation between
Ia afferents and joint angle with an average sensitivity close to the
experimental value. This suggests that the γscommand encodes
joint position information in the CNS with a parabolic θEP −γs
curve. Under the strategy of quadratic γscontrol, the peripheral
linear θEP −Ia relation could be maintained, and used to decode
actual angular information reliably from Ia afferents.
ACKNOWLEDGMENTS
Materials of this paper are based on the work supported by
a 973 basic research grant from the Ministry of Science and
Technology of China (No. 2011CB013304), a grant from the
Natural Science Foundation of China (No. 31070749) and a doc-
toral training grant from the Ministry of Education of China (No.
20100073110064).
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Conflict of Interest Statement: The
authors declare that the research
was conducted in the absence of any
commercial or financial relationships
that could be construed as a potential
conflict of interest.
Received:10April2012;accepted:13
August 2012; published online: 30 August
2012.
Citation: Lan N and He X (2012)
Fusimotor control of spindle sensitivity
regulates central and peripheral coding
of joint angles. Front. Comput. Neurosci.
6:66. doi: 10.3389/fncom.2012.00066
Copyright © 2012 Lan and He. This is
an open-access article distributed under
the terms of the Creative Commons
Attribution License,whichpermitsuse,
distribution and reproduction in other
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and source are credited and subject to any
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Frontiers in Computational Neuroscience www.frontiersin.org August 2012 | Volume 6 | Article 66 |12
Lan and He Central and peripheral coding of joint angles
APPENDIX
LIST OF ACRONYMS AND SYMBOLS
α—Alpha motoneurons
αd,αdyn—Alpha dynamic command
αs,αstat—Alpha static command
BS—Brachialis
Bsh—Biceps Brachii short head
γ—Gamma motoneurons
γd,γdyn—Gamma dynamic command
γs,γstat—Gamma static command
CNS—Central Nervous System
DOF—Degree of freedom
DP—Deltoid Posterior
ECRb—Extensor Carpi Radialis brevis
ED—Extensor Digitorum
EMG—Electromyography
θEP—Equilibrium point angle
Fm—Muscle force
GABA—γ-aminobutyric acid
GTO—Golgi Tendon Organ
Ia—Primary afferent from spindle
Ib—afferent from GTO
II—Secondary afferent from spindle
Lce—Muscle fascicle length
Lmt—Musculo-tendon length
PC—Pectoralis major Clavicle portion
pps—pulse per second
ROM—Range of Motion
SDN—Signal Dependent Noise
SNR—Substantia Nigra
Tlh—Triceps Brachii long head
Tlt—Triceps Brachii lateral head
VA — Vi r t u a l A r m
VM—Virtual Muscle
Frontiers in Computational Neuroscience www.frontiersin.org August 2012 | Volume 6 | Article 66 |13
